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## G = Q8×Dic5order 160 = 25·5

### Direct product of Q8 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Q8×Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Q8×Dic5
 Lower central C5 — C10 — Q8×Dic5
 Upper central C1 — C22 — C2×Q8

Generators and relations for Q8×Dic5
G = < a,b,c,d | a4=c10=1, b2=a2, d2=c5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 152 in 70 conjugacy classes, 51 normal (14 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C2×C10, C4×Q8, C2×Dic5, C2×Dic5, C2×C20, C5×Q8, C4×Dic5, C4⋊Dic5, Q8×C10, Q8×Dic5
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D5, C22×C4, C2×Q8, C4○D4, Dic5, D10, C4×Q8, C2×Dic5, C22×D5, Q8×D5, Q82D5, C22×Dic5, Q8×Dic5

Smallest permutation representation of Q8×Dic5
Regular action on 160 points
Generators in S160
(1 48 28 36)(2 49 29 37)(3 50 30 38)(4 41 21 39)(5 42 22 40)(6 43 23 31)(7 44 24 32)(8 45 25 33)(9 46 26 34)(10 47 27 35)(11 139 159 142)(12 140 160 143)(13 131 151 144)(14 132 152 145)(15 133 153 146)(16 134 154 147)(17 135 155 148)(18 136 156 149)(19 137 157 150)(20 138 158 141)(51 71 63 90)(52 72 64 81)(53 73 65 82)(54 74 66 83)(55 75 67 84)(56 76 68 85)(57 77 69 86)(58 78 70 87)(59 79 61 88)(60 80 62 89)(91 124 104 111)(92 125 105 112)(93 126 106 113)(94 127 107 114)(95 128 108 115)(96 129 109 116)(97 130 110 117)(98 121 101 118)(99 122 102 119)(100 123 103 120)
(1 68 28 56)(2 69 29 57)(3 70 30 58)(4 61 21 59)(5 62 22 60)(6 63 23 51)(7 64 24 52)(8 65 25 53)(9 66 26 54)(10 67 27 55)(11 122 159 119)(12 123 160 120)(13 124 151 111)(14 125 152 112)(15 126 153 113)(16 127 154 114)(17 128 155 115)(18 129 156 116)(19 130 157 117)(20 121 158 118)(31 90 43 71)(32 81 44 72)(33 82 45 73)(34 83 46 74)(35 84 47 75)(36 85 48 76)(37 86 49 77)(38 87 50 78)(39 88 41 79)(40 89 42 80)(91 144 104 131)(92 145 105 132)(93 146 106 133)(94 147 107 134)(95 148 108 135)(96 149 109 136)(97 150 110 137)(98 141 101 138)(99 142 102 139)(100 143 103 140)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(1 99 6 94)(2 98 7 93)(3 97 8 92)(4 96 9 91)(5 95 10 100)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)

G:=sub<Sym(160)| (1,48,28,36)(2,49,29,37)(3,50,30,38)(4,41,21,39)(5,42,22,40)(6,43,23,31)(7,44,24,32)(8,45,25,33)(9,46,26,34)(10,47,27,35)(11,139,159,142)(12,140,160,143)(13,131,151,144)(14,132,152,145)(15,133,153,146)(16,134,154,147)(17,135,155,148)(18,136,156,149)(19,137,157,150)(20,138,158,141)(51,71,63,90)(52,72,64,81)(53,73,65,82)(54,74,66,83)(55,75,67,84)(56,76,68,85)(57,77,69,86)(58,78,70,87)(59,79,61,88)(60,80,62,89)(91,124,104,111)(92,125,105,112)(93,126,106,113)(94,127,107,114)(95,128,108,115)(96,129,109,116)(97,130,110,117)(98,121,101,118)(99,122,102,119)(100,123,103,120), (1,68,28,56)(2,69,29,57)(3,70,30,58)(4,61,21,59)(5,62,22,60)(6,63,23,51)(7,64,24,52)(8,65,25,53)(9,66,26,54)(10,67,27,55)(11,122,159,119)(12,123,160,120)(13,124,151,111)(14,125,152,112)(15,126,153,113)(16,127,154,114)(17,128,155,115)(18,129,156,116)(19,130,157,117)(20,121,158,118)(31,90,43,71)(32,81,44,72)(33,82,45,73)(34,83,46,74)(35,84,47,75)(36,85,48,76)(37,86,49,77)(38,87,50,78)(39,88,41,79)(40,89,42,80)(91,144,104,131)(92,145,105,132)(93,146,106,133)(94,147,107,134)(95,148,108,135)(96,149,109,136)(97,150,110,137)(98,141,101,138)(99,142,102,139)(100,143,103,140), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,99,6,94)(2,98,7,93)(3,97,8,92)(4,96,9,91)(5,95,10,100)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155)>;

G:=Group( (1,48,28,36)(2,49,29,37)(3,50,30,38)(4,41,21,39)(5,42,22,40)(6,43,23,31)(7,44,24,32)(8,45,25,33)(9,46,26,34)(10,47,27,35)(11,139,159,142)(12,140,160,143)(13,131,151,144)(14,132,152,145)(15,133,153,146)(16,134,154,147)(17,135,155,148)(18,136,156,149)(19,137,157,150)(20,138,158,141)(51,71,63,90)(52,72,64,81)(53,73,65,82)(54,74,66,83)(55,75,67,84)(56,76,68,85)(57,77,69,86)(58,78,70,87)(59,79,61,88)(60,80,62,89)(91,124,104,111)(92,125,105,112)(93,126,106,113)(94,127,107,114)(95,128,108,115)(96,129,109,116)(97,130,110,117)(98,121,101,118)(99,122,102,119)(100,123,103,120), (1,68,28,56)(2,69,29,57)(3,70,30,58)(4,61,21,59)(5,62,22,60)(6,63,23,51)(7,64,24,52)(8,65,25,53)(9,66,26,54)(10,67,27,55)(11,122,159,119)(12,123,160,120)(13,124,151,111)(14,125,152,112)(15,126,153,113)(16,127,154,114)(17,128,155,115)(18,129,156,116)(19,130,157,117)(20,121,158,118)(31,90,43,71)(32,81,44,72)(33,82,45,73)(34,83,46,74)(35,84,47,75)(36,85,48,76)(37,86,49,77)(38,87,50,78)(39,88,41,79)(40,89,42,80)(91,144,104,131)(92,145,105,132)(93,146,106,133)(94,147,107,134)(95,148,108,135)(96,149,109,136)(97,150,110,137)(98,141,101,138)(99,142,102,139)(100,143,103,140), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,99,6,94)(2,98,7,93)(3,97,8,92)(4,96,9,91)(5,95,10,100)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155) );

G=PermutationGroup([[(1,48,28,36),(2,49,29,37),(3,50,30,38),(4,41,21,39),(5,42,22,40),(6,43,23,31),(7,44,24,32),(8,45,25,33),(9,46,26,34),(10,47,27,35),(11,139,159,142),(12,140,160,143),(13,131,151,144),(14,132,152,145),(15,133,153,146),(16,134,154,147),(17,135,155,148),(18,136,156,149),(19,137,157,150),(20,138,158,141),(51,71,63,90),(52,72,64,81),(53,73,65,82),(54,74,66,83),(55,75,67,84),(56,76,68,85),(57,77,69,86),(58,78,70,87),(59,79,61,88),(60,80,62,89),(91,124,104,111),(92,125,105,112),(93,126,106,113),(94,127,107,114),(95,128,108,115),(96,129,109,116),(97,130,110,117),(98,121,101,118),(99,122,102,119),(100,123,103,120)], [(1,68,28,56),(2,69,29,57),(3,70,30,58),(4,61,21,59),(5,62,22,60),(6,63,23,51),(7,64,24,52),(8,65,25,53),(9,66,26,54),(10,67,27,55),(11,122,159,119),(12,123,160,120),(13,124,151,111),(14,125,152,112),(15,126,153,113),(16,127,154,114),(17,128,155,115),(18,129,156,116),(19,130,157,117),(20,121,158,118),(31,90,43,71),(32,81,44,72),(33,82,45,73),(34,83,46,74),(35,84,47,75),(36,85,48,76),(37,86,49,77),(38,87,50,78),(39,88,41,79),(40,89,42,80),(91,144,104,131),(92,145,105,132),(93,146,106,133),(94,147,107,134),(95,148,108,135),(96,149,109,136),(97,150,110,137),(98,141,101,138),(99,142,102,139),(100,143,103,140)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(1,99,6,94),(2,98,7,93),(3,97,8,92),(4,96,9,91),(5,95,10,100),(11,90,16,85),(12,89,17,84),(13,88,18,83),(14,87,19,82),(15,86,20,81),(21,109,26,104),(22,108,27,103),(23,107,28,102),(24,106,29,101),(25,105,30,110),(31,114,36,119),(32,113,37,118),(33,112,38,117),(34,111,39,116),(35,120,40,115),(41,129,46,124),(42,128,47,123),(43,127,48,122),(44,126,49,121),(45,125,50,130),(51,134,56,139),(52,133,57,138),(53,132,58,137),(54,131,59,136),(55,140,60,135),(61,149,66,144),(62,148,67,143),(63,147,68,142),(64,146,69,141),(65,145,70,150),(71,154,76,159),(72,153,77,158),(73,152,78,157),(74,151,79,156),(75,160,80,155)]])

40 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 5A 5B 10A ··· 10F 20A ··· 20L order 1 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 5 5 5 5 10 ··· 10 2 2 2 ··· 2 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + - + + - - + image C1 C2 C2 C2 C4 Q8 D5 C4○D4 D10 Dic5 Q8×D5 Q8⋊2D5 kernel Q8×Dic5 C4×Dic5 C4⋊Dic5 Q8×C10 C5×Q8 Dic5 C2×Q8 C10 C2×C4 Q8 C2 C2 # reps 1 3 3 1 8 2 2 2 6 8 2 2

Matrix representation of Q8×Dic5 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 40 0
,
 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 20 38 0 0 0 38 21
,
 40 0 0 0 0 0 0 1 0 0 0 40 6 0 0 0 0 0 1 0 0 0 0 0 1
,
 32 0 0 0 0 0 16 25 0 0 0 39 25 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,1,0],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,20,38,0,0,0,38,21],[40,0,0,0,0,0,0,40,0,0,0,1,6,0,0,0,0,0,1,0,0,0,0,0,1],[32,0,0,0,0,0,16,39,0,0,0,25,25,0,0,0,0,0,1,0,0,0,0,0,1] >;

Q8×Dic5 in GAP, Magma, Sage, TeX

Q_8\times {\rm Dic}_5
% in TeX

G:=Group("Q8xDic5");
// GroupNames label

G:=SmallGroup(160,166);
// by ID

G=gap.SmallGroup(160,166);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,188,86,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^10=1,b^2=a^2,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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